Evaluating Plans through Restrictiveness and Resource Strength
Federico Pecora∗ , Amedeo Cesta
Institute for Cognitive Science and Technology
Italian National Research Council
name.surname@istc.cnr.it
Abstract
The planning and scheduling framework which is the object
of this paper consists in a loosely-coupled integration which
is achieved by cascading a classical planner and a general
purpose scheduler. The output of the planning phase yields a
partially ordered plan which is then integrated with time and
resource constraints to produce a scheduling problem. The
research described in this paper is aimed at understanding the
structural trademarks of the causal knowledge that a scheduling tool can inherit from STRIPS-based reasoners. Specifically, we analyze quality of the plans in terms of two wellknown properties of scheduling problems, namely the Restrictiveness and the Resource Strength. To this end, we describe a set of experiments carried out on a series of state-ofthe-art planners aimed at assessing the bias of different planning strategies in the context of the loosely-coupled framework.
Introduction
In recent years, increasing attention has been dedicated to
integrating Planning and Scheduling (P&S), with the aim
of extending the reach of automated solving to combinatorial optimization problems which require both causal and
time/resource reasoning. Integrated P&S frameworks are
typically obtained following two approaches. On one hand,
integrated P&S architectures can be designed using a “contextual” approach, so called because time/resources and the
causal sub-problem are dealt with contextually. This can be
achieved, as for instance in (Gerevini, Saetti, & Serina 2003;
Smith & Weld 1999; Currie & Tate 1991; Ghallab & Laruelle 1994), by expanding a strictly causal solving technique
with algorithms and data structures capable of dealing with
time and/or resources. Other contextual solvers adopt the
opposite strategy, namely starting from a scheduling-centric
approach and extending it with limited causal reasoning capabilities (Cesta, Fratini, & Oddi 2004; Jonsson et al. 2000;
Muscettola et al. 1992). The main drawback of contextual P&S integration strategies is that typically the “enhanced” system does not aquire a full set of capabilities.
∗
Also affiliated with the the Department of Systems and Informatics (DIS) of the University of Rome “La Sapienza”, Rome, Italy
— pecora@dis.uniroma1.it.
c 2005, American Association for Artificial IntelliCopyright °
gence (www.aaai.org). All rights reserved.
This is the case, for instance, of planners which support
PDDL2.1/2.2 (Fox & Long 2003; Edelkamp & Hoffmann
2004), an extension of the standard Planning Domain Definition Language (PDDL (Ghallab et al. 1998)) for modeling
actions with durations, a characteristic which does not alone
afford the versatility of common scheduling problem definition languages.
On the other hand, so-called “coupled” P&S consists
in linking separate implementations of the two solving
paradigms in order to achieve a bi-modular solver, in which
each component deals with the respective sub-problem. This
approach clearly presents some major difficulties since it
requires the definition of forms of information sharing between the two subsystems. Moreover, this type of integration can be achieved by a strong coupling of the two solving
components (i.e., in which every decision taken by one component is “propagated” by the other component), or, on the
other end of the spectrum, by loosely-coupling the two solving sub-systems (i.e., the planner produces a plan and the
scheduler then enforces the time and resource related constraints upon this plan). While a strongly-coupled architecture may seem to be more realistic, there is increasing evidence that often a loose-coupling of solving components is
more applicable to solve certain classes of real-world problems1 .
The P&S framework which is the object of our work consists in a loosely-coupled integration which is achieved by
cascading a classical planner and a general purpose scheduler. The output of the planning phase yields a partially ordered plan which is then integrated with time and resource
constraints. The resulting scheduling problem is then given
to the scheduler for resolution. The research described in
this paper stems from the necessity for an in-depth comprehension of some fundamental aspects related to this form
of integration. In particular, we address the issue of understanding the structural trademarks of the causal knowledge
1
This is the case, for instance, in military planning, where highranking officers take high-level planning decisions, disregarding
issues such as resource allocation or specific synchronization constraints, while the more scheduling-related decisions which must
be taken in order to make a plan operational occur afterwards.
An interesting discussion on this topic was brought up during the
ICAPS Workshop on Integrating Planning into Scheduling, June
2004.
that a scheduling tool can inherit from STRIPS-based reasoners. Thus, we set out to measure the structural properties
of the scheduling problems yielded from the planning phase
in the light of the subsequent scheduling. The present analysis extends the results obtained in (Pecora, Rasconi, & Cesta
2004; Cesta, Pecora, & Rasconi 2004), in which we focused on the bias produced by different planning approaches
(heuristic search and planning graph based) in the light of
makespan-optimizing scheduling. In this paper, our aim is
to assess the bias of the planning phase on two well-known
properties of scheduling problems, namely the Restrictiveness and the Resource Strength. These two structural properties of scheduling problems are commonly used2 as control parameters for the random generation of Resource Constrained Project Scheduling Problems with minimum and
maximum time lags (RCPSP/max) (Brucker et al. 1998).
To this end, we describe a set of experiments carried out
on a series of state-of-the-art planners aimed at generating
scheduling problems which exhibit varying degrees of these
two parameters.
This paper is organized as follows. We first present the
general experimental setup within which our analysis is carried out. To this end, we first describe more precisely the
loosely-coupled P&S framework (an architecture similar to
R EAL P LAN -MS (Srivastava 2000)) which our work focuses
on, as well as the benchmark set on which our analysis is
performed. We then describe the details of the structural
properties we wish to measure, while the section that follows
describes an extention to one of the planners analyzed in the
experiments aimed at increasing the detail of our observations. The main results of this paper consist in analyzing
and comparing the performance in terms of the measures introduced earlier of four state-of-the-art planners. The results
are then put together and further analyzed, and we conclude
with a summary of our findings as well as an outlook for
future work.
Experimental Setup
In an integrated P&S context, time and resource constraints
as well as causal dependencies are contemplated in the initial
problem definition. Every operator is inherently associated
to a time duration and requires a certain quantity of consumable multi-capacity resources that ensure its executability.
The general schema we use in this investigation is as follows (see figure 1). A causal model of the environment is
given as input to a planner, i.e., the domain representation
and the problem definition, both expressed in a STRIPS-like
formalism. This model does not contemplate time and resource related constraints, which are accommodated after
the planning procedure has taken place. In order to produce a problem specification which can be reasoned upon by
the scheduling procedure, the Partial Order Plan (POP) produced by the planner is integrated with time and resource
related information by means of a plan adaptation procedure. This procedure produces a minimal-constrained deordering (Bäckström 1998) of the POP and integrates it with
2
See for instance (Schwindt 1998).
POP
Planner
Info integration
& Deordering
completePOP π
Scheduler
schedule
Planning
problem
p
Time/Resource info
Durations and
Resource Usage
time
Causal model
PDDL Domain
Representation
PDDL Problem
Definition
Figure 1: The loosely-coupled reference framework, in which a
planning problem p is solved by the planner, ultimately yielding a
scheduling problem π.
time and resource related information to produce what we
shall call a completePOP. In more formal terms:
Definition 1. A completePOP P is a tuple
hT , P, R, C, U, Di where
• T = {T1 , . . . , Tn } ∪ {T0 , Tn+1 } is the set of tasks which
correspond to the n activities in the POP produced by the
planner; T0 and Tn+1 are called source and sink activities;
• P is a set of precedence constraints between the tasks,
where Ti ≺ Tj means that task Ti must be completed
before the execution of task Tj can begin;
• R = {R1 , . . . , Rm } is a set of renewable resources;
• C = {C1 , . . . , Cm } are the capacities of the resources;
• U : T × R → N ∪ {0} is the function which determines
the resource usage attributes of a task, that is, U(T, R) =
u if task T uses u units of resource R (u = 0 indicates
that T does not use resource R);
• D : T → N ∪ {0} is the function which determines the
durations of the tasks, that is, D(T ) = d if the duration of
task T is d time units.
According to the definition above, a completePOP coincides with a Resource Constrained Project Scheduling Problem (RCPSP) (Brucker et al. 1998) and can be visualized in
the form of a precedence graph.
A General Multi-Agent Problem Template
Our aim in this paper is to analyze the performance in terms
of plan quality of different planners within the above mentioned loosely-coupled framework. More specifically, we
are interested in assessing the influence of different planning
strategies on the structural characteristics of the scheduling
problems they lead to. This section briefly illustrates the
main features of the benchmark problems employed in the
experimental evaluation.
We focus on a multi-agent inspired planning domain
which produces completePOPs with high levels of concurrency by encapsulating the notion of executing agent. Concurrency is obtained by inducing the presence of multiple
agents in the POP. Having augmented the STRIPS subset
of the PDDL language with some simple extensions to allow the specification of durations and resource usage for the
operators (these directives are ignored by the planner and integrated into the POP to form the completePOP), the general
structure of a problem which leads to completePOPs which
exhibit high degrees of concurrency is shown in figure 2,
where :capacity 0 denotes an object which is not a resource. Given this structure, the number of agents in the
completePOP is determined by the number of objects with
non-zero capacity, i.e., |{A1, . . . , An}|, and the resource usage of each task is given by the :uses clause in the abstract
operator specification.
(define (domain . . . ) . . .
(:action op
:parameters (?a - agent . . . )
:precondition ( . . . )
:effect ( . . . )
:uses (?a USAGE)
:duration DUR) . . . )
(a)
(define (problem . . . ) (:domain . . . )
(:objects
A1, . . ., An - agent :capacity CAP
B1, . . ., Bm - type :capacity 0 . . . )
(:init . . . )
(:goal . . . ))
(b)
Figure 2: General structure of augmented PDDL domain (a) and
problem specifications (b) which lead to completePOPs with multiple agents. Bold typeface indicates directives which are ignored
by the planner.
A very simple example “instantiation” of this somewhat
general template could be the following: a team of robotic
agents can move from one room of an environment to another (so long as there is a door connecting the two adjacent
rooms), and they can perform a simple operation on certain types of objects (iff they are in the same room as
the object). In order to model the fact that agents do not
perform an operation on two objects at the same time,
each action :uses one unit of the resource agent. This
equates to modeling the agents as binary resources for the
scheduler. The idea is that, once the planning has taken place
(disregarding this resource usage information), the scheduler
will enforce the resource usage constraints, adding as a consequence a precedence constraint during the solving phase
which serializes two operations on the same object which
otherwise were independent.
Notice also that it is possible to model this binary re-
source constraint causally, by adding not (busy ?a)
to the preconditions and busy ?a to the effects of action operation. The busy predicate can be seen as a
“causal excuse” for the resource capacity constraint modeled with the :uses clause, and can be reset by a dummy
finish action (whose effect is not (busy ?a)). Notice that in this manner we would be producing scheduling
problems without resource contention peaks, thus transferring the burden of resolving these contentions to the planning phase. It has been shown in (Pecora & Cesta 2002;
R-Moreno et al. 2002) on a very similar domain that adopting this modeling strategy heavily affects the efficiency of
the planning procedure, making it very difficult to solve even
problems of small size.
Restrictiveness and Resource Constraints
There is a consolidated body of work in the field of scheduling which deals with classifying scheduling problems with
respect to various quantifiable properties. Our aim in this paper is to interpret these structural characteristics in the light
of different implementations of the planning phase which
constitutes the first component of the loosely-coupled framework. In this section we present the two well-known structural factors of scheduling problems which we will focus
on. In the remainder of this paper we will use these factors
to measure the quality of the scheduling problems obtained
with a number of state-of-the-art planners.
Order Strength. The first parameter we focus on quantifies the effect of the precedence constraints contained in the
plan (i.e., the causal structure of the scheduling problem) on
the possible execution sequences of the tasks. Our aim is to
measure to some extent the magnitude of the search space
which is explored by the scheduling phase. To this end, we
borrow the notion of restrictiveness from Project Scheduling.
Given a completePOP hT , P, R, C, Di, let l(P) be the
number of linear extensions of the causal structure determined by the precedence constraints in P. The restrictiveness of the completePOP3 can be defined as follows:
l(P)
|T |!
where |T |! represents the maximum number of linear extentions of a partial order in the set of tasks T . The restrictiveness measures the degree to which the precedences
between activities restrict the number of feasible execution
sequences of the tasks. For completely parallel causal structures, the restrictiveness is 0, while it is 1 for serial task networks. In general, the higher σ, the less different alternatives
exist to resolve resource conflicts between tasks (by defining additional precedence constraints). As a consequence,
the restrictiveness is directly linked to the hardness of many
RCPSP/max problems (both with respect to feasibility and
optimization) (Schwindt 1998).
σ = 1 − log
3
The restrictiveness of a directed graph is used as a control
parameter for the random generation of RCPSP/max problems,
e.g. (Schwindt 1998).
Unfortunately, calculating σ is #P-complete, since it involves counting linear extensions (Brightwell & Winkler
1991). As shown in (Mastor 1970), a good approximation
of the restrictiveness is the order strength. Given a completePOP π, this measure can be determined very efficiently
and is defined as follows:
¯ ¯
¯P ¯
OSπ =
(1)
|T | (|T | − 1) /2
i.e., the number of precedence relations, including the transitive ones (P denotes the set of precedence relations in the
transitive closure of the precedence graph), divided by the
theoretical maximum number of constraints. It is shown
in (De Reyck & Herroelen 1996) that OS outperforms other
significant network measures regarding the correlation with
the hardness of RCPSP/max instances.
The OS parameter thus expresses the properties of the
causal structure determined by the planning phase, and has
no connection with the resource constraints contained in the
problem. As we will see, different planning strategies yield
plans which differ in causal structure, thus the planning algorithms themselves are also connected to the hardness of
the resulting scheduling problems instances.
Resource Strength. One of the main features of schedulers consists in the capability to deal with resources in addition to complex time-constraints. In order to understand
the bias produces by different planning strategies on the
resource-related characteristics of the resulting scheduling
problems, we employ another well-known measure, namely
the resource strength4 . Given the k-th resource in the comk
pletePOP, we denote with rmin
the maximum usage of this
resource by a single task in the completePOP, that is
k
rmin
= max U(T, Rk )
T ∈T
is resource-feasible. We call a problem in which Rk adheres to the first requirement k-parsimonious. Notice also
k
is a lower bound for Ck , since it represents the
that rmin
maximum usage of resource Rk by a single task in the completePOP. Therefore, in a k-parsimonious completePOP we
k
k
have that Ck is bounded in the interval [rmin
, rmax
], since
these bounds represent, respectively, the smallest and largest
capacity which can be given to Rk in order for the scheduling problem to be resource feasible with respect to Rk . If
this is the case, then clearly RSk is also bounded in the interval [0, 1], while RSk may be greater than 1 of the completePOP is not k-parsimonious.
In conclusion, the RS parameter takes into account the
resource constraints which are given by the requirements of
the tasks on one hand, and the limited resource capacities
on the other, thus it measures the scarcity of the resource
availability with respect to the requirements. In the following sections, our analysis focuses on the average RS over
the resources which are employed in the completePOP, that
is
P
RSk
RS π = k:Rk ∈Ru
(2)
|Ru |
where Ru ⊆ R is the set of used resources, i.e., for which
U(T, R) > 0 for some T ∈ T . Notice that a completePOP
may not prescribe the use of all resources modeled in the
original problem, since the particular action instantiations
which achieve the goal (the POP produced by the planner)
are obtained by means of a purely causal deduction, and
as such does not explicitly take into account any resourcerelated metric. Indeed, some among the planners we analyze in the following sections over-commit more than others
with respect to resource utilization. This reflects strongly on
the average resource strength: the higher RS π , the less constraining are the capacities of the resources with respect to
the task network.
k
rmax
Also, let
denote the peak demand of resource Rk in the
precedence-preserving earliest start schedule (i.e., the infinite capacity solution). The resource strength of resource
Rk is thus defined as
RSk =
k
Ck − rmin
k
k
− rmin
rmax
where Ck ∈ C is the capacity of resource Rk . The resource strength expresses the relationship between the resource demand and the resource availability in the given
completePOP.
Notice that a scheduling problem in which the capacity
k
Ck of the k-th resource is at most rmax
(i.e., the maximum
potential claim of resource k in the precedence-preserving
earliest start schedule) is such that resource k is “welldimensioned”. Conversely, if Ck exceeds the peak demand
of Rk , then this resource’s capacity is unnecessarily large,
k
since rmax
is an upper bound on Ck such that the problem
4
This parameter was first defined in (Cooper 1976; AlvarezValdez & Tamarit 1989). The definition we use here is taken
from (Kolisch, Sprecher, & Drexl 1995).
While the scheduling literature points to numerous other parameters which could be useful in the context of this investigation, some of which we will be analyzing in future work,
in the following section we observe how the planning phase
affects the two properties defined above.
As a final note, it is interesting to notice that taking into
account “classical” properties of project scheduling problems rather than the strong-coupling constraint density we
have defined in our previous work, allows us to relax the restricting assumptions we have made with respect to resource
capacities and usage. In particular, parameters such as the
two we analyze in this work are defined for any causal structure in the scheduling problem, as well as for multi-capacity
resources and multiple-resource usage constraints.
Enhancing B LACK B OX for Obtaining Multiple
Causal Solutions
Before describing in detail the structural characteristics of
the scheduling problems which we want to observe, we describe an extention to the loosely-coupled framework aimed
at enhancing the detail of our observations. Among the planners used in our experiments, we employ a planning subsystem which is capable of producing multiple plans, thus
yielding, after the information integration and de-ordering
procedure has taken place, multiple completePOPs. Having
more solutions to the causal part of the P&S problem provides us with a richer set of scheduling problems on which
to perform the structural analysis by means of the measures
defined previously. This extends the results obtained in (Pecora, Rasconi, & Cesta 2004; Cesta, Pecora, & Rasconi 2004)
because it enables us to observe how not only different planning strategies but also different methods for extracting solutions within a certain planning strategy can influence the
structure of the underlying scheduling problem. This is particularly true in the case of planning graph based planning,
in which a planning graph contains many valid plans. Moreover, these plans are partitioned into sets according to the
level of expansion at which they are obtained. In other
words, given the minimum level l at which there exists a
solution to the planning problem, instead of obtaining one
scheduling problem π l , we obtain the set of scheduling problems {π1l , π2l , . . .} which derive from the solution subgraphs
of the planning graph5 .
In order to obtain multiple completePOPs in a single
level of planning graph expansion, we have implemented
a modified version of the B LACK B OX planning system,
which we call M B LACK B OX (for multiple-B LACK B OX).
This enhancement is obtained by replacing the Z C HAFF
SAT-solver (Moskewicz et al. 2001) originally integrated
within the B LACK B OX system with M C HAFF, an enumerative variation of Z C HAFF (Moskewicz et al. 2001;
Moskewicz 2004), i.e., which is capable of finding multiple models of a given well-formed formula (see figure 3).
The experiments described in the following sections thus fo{POPl1 , POPl2 , . . .}
G RAPHPLAN
graph Gl
subgraphs {Gl1 , Gl2 , . . .}
Graph2WFF
WFF
models {φ1 , φ2 , . . .}
Info integration
& Deordering
M C HAFF
{π1l , π2l , . . .}
Figure 3: Enhancing B LACK B OX with enumerative capabilities
by replacing Z C HAFF with M C HAFF. Gl denotes the planning
graph expanded up to level l.
5
Moreover, by increasing the level of expansion of the planning graph we can produce the sets of scheduling problems
{π1l , π2l , . . .}, {π1l+1 , π2l+1 , . . .}, . . . , {π1m , π2m , . . .}, where m is
the index of planning graph level-off. The analysis of OS and RS
on further levels of expansion of the planning graph is outside the
scope of this paper, and will be addressed in future work.
cus also on the variation of OS and RS within the sets of
solutions produced by M B LACK B OX.
Experiments
The two parameters shown above represent attributes of
scheduling problems which are widely accepted as meaningful. In the context of this investigation, OS characterizes the causal structure of planner-derived scheduling problems, measuring the restrictiveness of the task network determined by the planning phase, thus indicating the “degree
of freedom” available to the scheduler in resolving conflicts.
RS on the other hand quantifies the constrainedness of completePOPs with respect to the resources allocation determined by the planner. Both OS and RS provide a means
to interpret the effect of the decisions taken by the planner
with respect to the causal and resource allocation problem
represented by the input planning problem.
Benchmark Problems
For the purpose of the experiments described in this section, the general multi-agent domain definition described
earlier was instantiated according to the dependency graph
shown in figure 4. The domain consists in a total of six
operators, each requiring an object as well as a certain
type of agent (type one or two). Moreover, each operator
:uses a particular amount of the executing agent, and has
a given :duration. Our benchmark consists of 100 randomly
this domain
¤ ®in the form
£
­ generated
£ problems ¤within
p = A1 , A2 , C1min , C1max , C2min , C2max , O , where
• A1 (A2 ) is the number of agents of type one (two);
¤
¤ £
£
• C1min , C1max ( C2min , C2max ) is an interval within
which lies the :capacity of every agent of type one
(two);
• O is the number of objects (i.e., with :capacity 0).
Each problem was generated by randomly choosing A1 , A2 ,
O, as well as the capacity of each agent of type one and two
within the corresponding interval. Un-resolvable problems
were avoided by ensuring that the C1min ≥ 4 and C2min ≥ 3.
Notice that the absence of negative effects in this domain make the corresponding planning problems polynomial. This choice is determined by our interest in aspects
related to the solution quality rather than the performance
of the planners we consider. To this end, the 100 generated
benchmark problems are intended to provide an easy set of
problem instances from the planning point of view. Conversely, the additional resource and time related information poses a more elaborate scheduling sub-problem, with
respect to which the tested planners exhibit a varying degree
of commitment.
As we will see in the experimental analysis, the regularity
of this domain introduces a strong dependency of the results
obtained by the various planners on the size of the problem
instance. In addition, notice that the domain is such that we
expect to obtain little variation in the causal structure of the
plans obtained with different planners6 . Nonetheless, as we
6
This domain can be interpreted as a workflow-management application, in which there is no real difficulty in determining the se-
start
2
a2
2
a1
op12
o
op11
o
P&S problem.
The same set of 100 benchmark problems were solved
by FF (Hoffmann & Nebel 2001), LPG (Gerevini, Saetti,
& Serina 2003) and CPT (Vidal & Geffner 2004), and OSπi
and RS πi were also computed for the 100 solutions obtained
from these planners.
Resources
eff1(o)
a2
3
a2
We begin by analyzing the resource related aspects of the
obtained completePOPs. Figure 5 shows the value of RS
for the various planners considered. In particular, only the
first plan produced by M B LACK B OX is considered (i.e., the
original non-enumerative version of B LACK B OX). Recall
2
op3
o
op2
o
1.4
M B LACK B OX
eff3(o)
eff2(o)
4
i
LPG RS πi
1
a1
RS π0
FF RS πi
1.2
CPT RS πi
0.8
op4
0.6
o
0.4
eff4(o)
0.2
a1
2
0
op5
o
20
30
40
50
60
70
80
90
100
(M B LACK B OX in non-enumerative mode).
Figure 4: The dependency graph of the multi-agent domain used
for the generation of the benchmark set. Ovals represent predicates
and rectangles represent operators.
will see the relative regularity of the benchmark set does not
affect the generality of our observations.
From POPs to Scheduling Problems
To begin with, the benchmark problems described above
were solved using M B LACK B OX. As mentioned, this planner employs M C HAFF in order to produce multiple solutions
to a single planning problem. On the 100 benchmark problems, M B LACK B OX yields a total of 7046 plans (the enumeration performed by M C HAFF yields between 10 and 200
unique plans for each problem). Each solution represents a
completePOP πij , where i ∈ [1, 100] and j refers to the j-th
unique solution to problem i. We then calculate OSπj and
i
RS πj . Moreover, we extract from these measures OSπmax
,
i
max
10
Figure 5: Average resource strength for the various planners
eff5(o)
i
0
min
OSπmin
, RS πi and RS πi , which represent the maximum
i
and minimum values of OS and RS obtained for the i-th
quence of operators which achieves a goal, and in which the planner’s task is essentially that of deciding an allocation of agents to
operations.
that high values of RS indicate that the corresponding plan
is under-constrained with respect to resource capacity constraints, thus that the resources are assigned in such a way
that their capacity limitations are less likely to entail resource conflicts. It is immediate to notice the gap in performance between FF and CPT on one hand, and M B LACK B OX and LPG on the other. In most problems the former
planners employ only one or two agents out of the overall pool of available agents. The strongly sub-optimal allocation of resources (agents) to tasks clearly has no impact on the quality of the solution to the causal sub-problem
(the POP), while it heavily affects the quality of the completePOP as a whole. The scheduling problems produced
by FF and CPT are thus very constrained from the point of
view of resources. Overall, this may have some disadvantages with respect to the scheduling phase, such as longer
makespans, less robust solutions, and so on.
The poorer quality of the plans obtained with FF and CPT
demonstrates the fact that the over-committing nature of the
performance-oriented heuristics employed by these planners
appears to be counter productive in the light of the subsequent scheduling phase. Conversely, B LACK B OX and LPG
have the nice property of committing less to resource peak
leveling decisions, thus maintaining a lower level of invasiveness in the decision space of the scheduler. This phenomenon is even more interesting in the light of the fact that
both B LACK B OX and LPG are (though in different ways)
planning graph based planners. These results thus point to
an interesting load balancing quality of the planning graph
paradigm.
Let us now observe the results obtained with the enumerative planner M B LACK B OX. As shown in figure 5, the performance of LPG is in line with that of the classical singleplan B LACK B OX, reaching for many problems higher peaks.
Conversely, plotting the results obtained with the enumerative variant M B LACK B OX yields the results shown in figure 6. These results show that extracting more solutions
0.3
M B LACK B OX
OSπ0
FF OSπi
LPG OSπi
CPT OSπi
0.25
i
0.2
0.15
0.1
1.6
M B LACK B OX
1.4
max
RS π
LPG RS πi
0.05
i
1.2
0
10
20
30
40
50
60
70
80
90
100
Figure 7: Order strength for the the various planners ( M B LACK B OX in non-enumerative mode).
1
0.8
eration of the benchmark problems. In fact, the domain in
question yields planning problems whose solutions have a
structure which depends strongly on the size of the problem
(see figure 8). In addition to the plateau effect, the results
0.6
0.4
0.2
0
24
0
10
20
30
40
50
60
70
80
90
100
Figure 6: Average resource strength for LPG and the best performance of M B LACK B OX.
20
to a given planning problem affords us a choice between
“equivalent” completePOPs with potentially very different
characteristics from the point of view of resource allocation.
In conclusion, notice that the enumeration of solutions performed by M C HAFF also ignores the resource related information in the problem instance. Indeed, these results suggest
the possibility to equip the SAT solver with a more informed
heuristic for solution space exploration that takes into account resource related metrics.
16
Causal Structure
We now turn our attention to the causal characteristics of
the POPs produced by the planners considered above. As
described previously, the output if the planner consists in a
sequence of tasks which are then de-ordered to produce the
precedence graph hT , Pi. The structural characteristics of
this graph affect the scheduling phase in a number of ways,
and, as noted earlier, a good predictor for the hardness of
a scheduling problem is represented by the restrictiveness,
which can be in turn approximated by OS.
Figure 7 shows the values of OS obtained with the four
planners considered in the previous section. Again, we start
by analyzing M B LACK B OX in non-enumerative mode. We
can immediately notice three characteristics of the results:
the presence of plateaus, the fact that OSπi for all planners
except M B LACK B OX coincide on all problems, and that the
values of OS appear to be scaled. These phenomena are due
to the particular structure of the domain used to produce the
benchmark problems.
The first two aspects can be easily explained in terms of
the particular structure of the domain employed for the gen-
M B LACK B OX OSπ 0 × 100
i
|O|
22
18
14
12
10
8
6
4
0
10
20
30
40
50
60
70
80
90
100
Figure 8: Dependency of the order strength on problem size in the
benchmark set.
show that LPG, CPT and FF all tend to produce very similar solutions, so similar that from the strict point of view
of the causal structure (i.e., not considering how resources
are allocated) the plans are identical. Again, this is a direct
consequence of the “simplicity” (from the planning point
of view) of the domain. Indeed, further experiments show
that this does not occur on more complex planning problems (such as the planning competition benchmarks), where
the three planners yield solutions with values of OS that are
not as easily comparable. This observation indicates that the
benchmark problems considered in this paper yield a search
space whose topology points all three heuristics to solutions
with the same structural characteristics. Indeed, the simplicity of the domain is also the reason for the apparent scaling
between the values obtained with B LACK B OX and those obtained with the other three planners. In fact, experiments on
more complex domains do not yield scaled values of OS.
However, given that the benchmark set is so regular, it
is somewhat unexpected that B LACK B OX obtains solutions
with a different causal structure. These results seem to emphasize the difference between the SAT-solver based plan
extraction mechanism and the search procedures employed
by the other planners. The fact that the value of OS is lower
for B LACK B OX than for the other planners confirms the results obtained in (Pecora, Rasconi, & Cesta 2004), in which
we have shown, by means of a different experimental analysis, that planning graph based planners tend to produce
scheduling problems which are easier than those produced
by planners based on heuristic search (recall that OS is an
indicator of the hardness of a scheduling problem).
A natural question at this point is the following: how does
the enumeration of solutions performed by M B LACK B OX
affect OS? Figure 9 shows the highest (worst) and lowest (best) order strength of the multiple solutions obtained
by M B LACK B OX. An interesting aspect of these results is
0.3
min
OSπ
i
max
OSπ
i
FF, LPG, CPT OS
M B LACK B OX
M B LACK B OX
0.25
0.2
0.15
0.1
0.05
0
10
20
30
40
50
60
70
80
90
100
Figure 9: Order strength for the the various planners, with best
(OSπmin
) and worst (OSπmax
) performance of M B LACK B OX.
i
i
that the best values of OS obtained by M B LACK B OX are
also the first, i.e., OSπmin
= OSπi0 , ∀i. Recall that with
i
respect to the quality of resource allocation (RS), we find
that exploring different solutions to the planning problem is
advantageous (see figures 5 and 6 in the previous section).
Conversely, exploring further solutions does not yield an improvement with respect to the order strength, which is also
increasing (i.e., there exist less different alternatives to resolve resource conflicts). This is to be expected: increasing
the resource strength makes the scheduling problems more
robust with respect to resource failures, but decreases the
ease with which the scheduler can resolve resource conflicts.
Further Analysis of the Results
The experimental results described in this paper allow us to
draw some interesting conclusions on how different planning strategies can benefit loosely-coupled P&S.
Planners as Resource Allocators. First, we have observed the characteristics of the various planning strategies
on resource allocation by analyzing the average resource
strength of the scheduling problems which result from the
planning procedure. In this context we have seen how two of
the most representative heuristic search based planners perform poorly with respect to resource allocation. Conversely,
B LACK B OX and LPG, which base the search space representation on planning graphs, achieve much better results.
The results point to the fact that planning graph based representations retain the intrinsic quality of favoring the extraction of plans with a higher degree of load balancing. Dually,
the search strategies employed by FF7 and CPT favor the
exploration of states achieved by means of a minimal use
of resources for execution. Notice that while it is true that
resource capacities are ignored by the planning phase, the
presence of resources for execution is not. There is no real
“reason” for a planner to prefer the use of many agents rather
than the strict amount necessary for causal satisfiability. Our
results show that nonetheless, the planning graph data structure, which is a compact representation of the entire search
space save the states pruned by mutex propagation, allows
planners such as B LACK B OX and LPG to “inadvertently”
provide solutions which make use of a greater number of resources for execution, thus effectively doing a better job at
load balancing.
It is interesting to notice that a further uninformed exploration of the solution space achieved with the enumerative
SAT-solver M C HAFF improves these results further. This
makes a case for employing an enumerative planner for the
purposes of loosely-coupled P&S, in which it is better to
have more “equivalent” scheduling problems in the presence
of invalidating run-time conditions.
The Bias of Planning on Restrictiveness. Our interest in
the restrictiveness of the scheduling problems produced by
the planners analyzed in this paper lies in the fact that it represents a way to put a number on the causal structure of the
precedence graph which constitutes the scheduling problem.
Also, measuring the restrictiveness is interesting because
this measure quantifies the degree to which the planner restrains the search space of the resulting scheduling problem.
If a planner obtains higher degrees of restrictiveness compared to another planner on the same problems, then this is
an indication that the first planner invades the decision space
of the scheduler more than the latter. Our analysis has shown
that B LACK B OX retains the capability of guiding its search
procedure towards a solution with lower degrees of restrictiveness, thus maintaining a lower commitment with respect
to the subsequent scheduling phase. These result confirm
a previous analysis (Pecora, Rasconi, & Cesta 2004) based
on different structural properties of the scheduling problem,
but which relied on the assumption that all resources were
binary (Ci = 1, ∀i). The use of OS and RS has allowed us
to relax this assumption.
The Tradeoff Between Robustness and Resource Efficiency. The results obtained by means of the enumera7
Notice that FF employs relaxed planning graph propagation to
compute the heuristic estimator. Nonetheless, the heuristic value
so obtained represents only an estimate of goal distance, and does
not incorporate any other information derivable from the planning
graph representation.
tive planner M B LACK B OX show that exploring the neighborhood of the fist satisfying assignment yields plans with
higher degrees of restrictiveness (higher values of OS), but
also improved performance with respect to resource capacity limitations (higher values of RS). Indeed, this does not
come as a surprise. In fact, it is reasonable to expect a tradeoff between these two characteristics of the completePOPs.
On one hand, if the precedences between the tasks in the
scheduling problem allow a high number of valid execution
sequences (i.e., low OS), then it is reasonable to expect that
the only way to safeguard against solutions which are fragile with respect to resource failures is to obtain a constraint
network with higher restrictiveness in order to exclude these
fragile solutions. Dually, it is straightforward to see that in
order to increase the number of solutions to a P&S problem
it is necessary to revert to a completePOP which potentially
forebodes more resource contention peaks (lower RS), but
at the same time allows the scheduler more degrees of freedom for computing a solution which makes efficient use of
the resources. In the light of these considerations, we can
see high restrictiveness (and resource strength) as a symptom of robustness, while low restrictiveness (which entails
low resource strength) implies fragility.
It is important to keep in mind that in the context of
loosely-coupled P&S, OS and RS represent characteristics
of alternative causal solutions to a given P&S problem (i.e.,
alternative scheduling problems). We have shown that it
is possible to obtain a multitude of “equivalent” scheduling
problems (using different planners or an enumerative planner). If we can choose, within the context of one P&S problem, which scheduling problem to schedule for, the considerations made above become quite important. Depending,
for instance, on the projected execution scenario, it may by
useful to immediately select the scheduling problem which
exhibits the appropriate tradeoff between (low) restrictiveness and (high) resource strength.
Conclusions and Future Work
The work described in this paper stems from the general
question of how to extend a strictly causal solver with time
and resource reasoning capabilities. In this context, we focus on a loosely-coupled P&S integration, that is, a framework in which the output of a classical planner is piped into
a scheduler. This approach to P&S integration is suited for
applicative contexts which require scheduling decisions to
occur without the possibility of intervening on the planning
decisions, which have already taken place and are irrevocable. Notice also that this component-driven approach is
attractive because of its ease of implementation using offthe-shelf general purpose tools. In this context, the present
investigation, which extends the results obtained in (Pecora, Rasconi, & Cesta 2004), is aimed at assessing the pros
and cons of different planning strategies within the looselycoupled framework.
The results described in this paper provide an assessment
of how suited different planning strategies are with respect
to two important characteristics of the resulting scheduling
problems. We have shown how restrictiveness and resource
strength, two well-known attributes of scheduling problems,
provide novel and interesting quality metrics for plan quality. The results of these measurements have exposed new
strengths and weaknesses of classical planners, which complement commonly accepted criteria for plan quality such as
those employed in the planning competition, and may point
towards new application scenarios for current AI planning
technology.
We believe that the issues put forth in this paper forebode
interesting new directions of research for planning, such as
new planning strategies which explicitly take into account
the measures described in this paper, as well as investigating
other applicable measures for plan quality.
Acknowledgments. This research is partially supported
by MIUR (Italian Ministry of Education, University and Research) under project ROBO C ARE: “A Multi-Agent System
with Intelligent Fixed and Mobile Robotic Components”,
L. 449/97 (http://robocare.istc.cnr.it). The
Authors wish to thank Riccardo Rasconi and Simone Fratini
for having fostered interesting common research.
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